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Workshop in Financial Engineering The Stock Game Dr. J. Ren Villalobos, Joel Polanco and Marco A. Gutierrez Industrial Engineering Dept. Arizona State University

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Introduction This game was originally prepared for the IIE regional student chapter conference We will use it to start the discussion on stock selection

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Objective Through hands-on exercises show the use of classical Industrial Engineering tools in devising Investment Strategies (Financial Engineering)

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Financial Engineering Financial Engineering is becoming a very hot topic in Industrial Engineering Programs It has been established as a specialization area in programs such as: Georgia Tech, Columbia and Michigan

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What is Financial Engineering? Combining or carving up existing instruments to create new financial products. (http://www.duke.edu/~charvey/Classes/wpg/glossary.htm) Financial engineering is the application of mathematical tools commonly used in physics and engineering to financial problems, especially the pricing and hedging of derivative instruments. (from Neil D. Pearson, Assoc. Prof. UIUC) Financial engineering is the use of financial instruments such as forwards, futures, swaps, options, and related products to restructure or rearrange cash flows in order to achieve particular financial goals, particularly the management of financial risk. (www.stanford.edu/~japrimbs/00- Intro.ppt)

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Investment Game You will be given $100,000 of virtual money to invest in 7 stocks and a risk-free alternative which yields an interest of 2% per period You can divide up the money any way you want among the 7 stocks and the risk-free investment You will have the opportunity to change your original investment two times before the end of the game Assume that you borrowed the original $100,000 and that you have to repay it by making two payments of $55,000 at the end of rounds 2 and 3. Your job is to apply the IE (or other engineering) tools that you know and your intuition to maximize the value of your investment when the game is over. If you cant meet the scheduled payments your company goes bankrupt and the game is over for your team

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Example First Round: You are given in Excel the daily closing quote for two stocks for around two years.The information looks like:

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Example From the previous information, the closing price for stocks A (the stock in blue) and B (the stock in pink) at May 31, 2001 are: $8.55 and $25.64 respectively. Based on the information provided we make our investment allocation decision as $50,000 in stock A and $50,000 in Stock B, which corresponds to 5848 shares of Stock A and 1950 shares of stock B. Then you will be given information for the second period

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Example

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Your previous investment is now worth ($153,441.00, $76374 for Stock A and $77067 for stock B). Since stock B performed better than Stock A you decide to sell 4211 of your shares of stock A and use the proceedings to pay the $55,000 loan payment Now you are left with 1636 shares of stock A and 1950 of stock B and an investment value of $98,441.00 The information for the next period looks as follows

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Example

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The value of your stock of 1636 shares of stock A and 1950 of stock B is now $66223 Since you almost lost your shirt in the last period because of Stock B you now sell ALL your shares of this stock and 974 shares of stock A to pay your final loan payment. You are left only with 662 shares of stock A and none of stock B with a total market value of $11223 The information for the last period looks as follows

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Example

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Final Value Shares of Stock A= 662, Price = $19.14 Total = $12673 Shares of Stock B= 0, Price = $24.4 Total = $0 We made $12,673 but almost when bankrupt in the process Is there something that we can do to avoid going bankrupt and maximizing the return on the original investment?

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Your Turn to play You have borrowed $100,000 of virtual money to invest in 7 stocks and a risk-free alternative which yields an interest of 2% per period You can divide up the money any way you want among the 7 stocks and the risk-free investment you will be given forms to report the percentage of the total investment that you will allocate to each alternative You will have the opportunity to change your original investment two times before the end of the game You have to repay the loan by making two payments of $55,000 at the end of rounds 2 and 3. Your job is to apply the IE (or other engineering) tools that you know and your intuition to maximize the value of your investment when the game is over. If you cant meet the scheduled payments your company goes bankrupt and the game is over for your team

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Round 1 You have been provided with data for the seven stocks in the file: stockgame_set1.xls.

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First allocation You have seven minutes to analyze the data and decide what you will split your investment fund. You need to turn in the investment form with your allocation. Example of form

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Round 2 You have been provided with data for the seven stocks in the file: stockgame_set2.xls.

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Second allocation You have seven minutes to analyze the data and decide what you will split your investment fund. You need to turn in the investment form with your allocation.

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Results of the first allocation The current positions for the teams (after the first allocation) are:

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Round 3 You have been provided with data for the seven stocks in the file: stockgame_set3.xls.

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Third allocation You have seven minutes to analyze the data and decide what you will split your investment fund. You need to turn in the investment form with your allocation.

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Results of the Second allocation The current positions for the teams (after the secondallocation) are:

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Last period You have been provided with the stock prices data for the seven stocks after the last allocation in the file: stockgame_set4.xls.

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Game Results of the Second allocation The final positions in the competition are: The overall winner is Team

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Analysis Some of the teams used past price behavior (mean and standard deviation) to determine how to invest the funds. Is there something else that we can do? What about if we look at the return on investments that each one of the stocks gives? The information used for the game came from the closing prices for seven stocks: AA (A), IBM(B), INTC(C), JNJ(D), MCD(E),YHOO(F),THC(G)

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Analysis

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The stocks giving the most return tend to have the highest variation (as measured by the standard deviation) Ideally we want the average return to be as high as possible and the Std. Dev as small as possible

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Mean-Standard Deviation Diagram

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Analysis Can we reduce the risk (std. Deviation) by having a mix (portfolio) of stocks. What do we know. From Probability we know that: Let R = w 1 R 1 + w 2 R 2, Where R is the return of a portfolio with two securities having an average return of the returns of R 1, R 2 and weight of w 1, w 2 each (R 1 + R 2 =1)

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Portfolio Theory Let r be a portfolio with n assets, each with (random) rates of return of r 1, r 2, , r n and with expected values (means) of and standard deviations of, and weights Then Mean return of a portfolio The variance of a portfolio is Where is the covariance of assets i and j.

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Portfolio Theory The previous formula tells us the the variance of the portfolio is generally lower than the variance of all the individual components the more assets in the portfolio the less the variance. To better appreciate this suppose that you build a portfolio with equal weights of n independent assets with the same variance and mean. Then: The more assets the put in the portfolio the less the variability: If we put an infinite number of assets the variability would be zero and we would know with absolute certainty the value of the return of the portfolio

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Two Asset Portfolio The previous formula was for uncorrelated, equal weights, equal means and variances stocks For a two-asset portfolio suppose that the weight of asset 2 is . We want to know the relationship between the mean and the standard of the portfolio and the original two assets The mean and the variance of the portfolio are For (correlation) = 1 For (correlation) = -1 The portfolio variance is zero at

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Graphical Representation

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Three Stock Portfolio

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The efficient frontier

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Markowitz Model To find a minimum variance portfolio we fix the mean value at some arbitrary point r. Then we we find the feasible portfolio of minimum variance that has this mean. The formulation of the problem is as follows: This problem is solved by using Lagrangian Multipliers At the end we solve a series of equations to find the weights (or amount to invest) of the assets in a portfolio

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Solution The Markowitz selection portfolio portfolio has a quadratic objective function subject to linear constraints. This kind of problems can be solved by using Lagrangian multipliers as follows: L is known as the Lagrangian of the original problem. By differentiating L with respect to the decision variables, in this case the weights of the assets in the portfolio and setting these equations to zero and using the original linear constraints. This results in a series of linear equations that when solved they give the optimal portfolio If shorting is not allowed (the weights are not allowed to be negative), quadratic programming needs to be used. For simple cases Excel could be used

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Example for two stocks shorting allowed In this case we have four linear equations with four unknowns Solving this systems we get the weights of the portfolio

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Example Find the right amount to invest between the stocks of Intel ( 1 = 0.16103, 1 = 0.4151) and IBM ( 2 = 0.11791, 2 = 0.2388, 12 = 0.1032) if you have 100k to invest and the targeted return is a semi-annual 13%

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Solution Solving the system we get: w1 = 0.28, w2 = 0.72)

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Application to the stock game If we use the data for daily return, std. deviation, covariance and desired return of 13%, solving the system of equations that these values produced we get (for the first investment): This Negative value corresponds to selling short a stock, if we do not want to get negative values we need to use a technique called Quadratic Programming instead of doing this we set this value to zero and rebalance the solution

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Application to the stock game For the second and third round we get: The results for the previous strategy would be: The average return per period was about 14.7%

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How this results compares to the results of the other teams

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For Further reading Investment Science by David G. Luenberger